Subgroups — C₂₃×#18606 #19166232

Order 5888.

984 conjugacy classes of subgroups.

The subgroup lattice has 786 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3517	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3251	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3184	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3035	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1654	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1653	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1642	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1641	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1626	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1625	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1621	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1621	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1547	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1489	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1483	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#135	—	2	1	no	—
17	C₂₃×#557	C₂²	1	1	yes	characteristic
18	C₂₃×#553	—	2	2	no	—
19	C₂₃×#553	—	2	1	no	—
20	C₂₃×#550	—	2	1	no	—
21	C₂₃×#550	—	2	1	no	—
22	C₂₃×#550	C₂²	1	1	yes	characteristic
23	C₂₃×#550	C₂²	1	1	yes	characteristic
24	C₂₃×#547	—	2	1	no	—
25	C₂₃×#547	C₂²	1	1	yes	characteristic
26	C₂₃×#540	—	2	2	no	—
27	C₂₃×#540	—	2	1	no	—
28	C₂₃×#532	—	2	1	no	—
29	C₂₃×#532	C₂²	1	1	yes	characteristic
30	C₂₃×#532	C₂²	1	1	yes	characteristic
31	C₂₃×#529	—	2	1	no	—
32	C₂₃×#529	C₂²	1	1	yes	characteristic
33	C₂₃×#520	—	2	1	no	—
34	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#498	C₂²	1	1	yes	characteristic
36	C₂₃×#493	C₂²	1	1	yes	characteristic
37	C₂₃×#486	C₂²	1	1	yes	characteristic
38	C₂₃×#486	C₂²	1	1	yes	characteristic
39	C₂₃×#478	—	2	1	no	—
40	C₂₃×#475	C₂²	1	1	yes	characteristic
41	C₂₃×#474	C₂²	1	1	yes	characteristic
42	C₂₃×#469	—	2	2	no	—
43	C₂₃×#469	—	2	1	no	—
44	C₂₃×#468	—	2	1	no	—
45	C₂₃×#464	—	2	1	no	—
46	C₂₃×#463	—	2	1	no	—
47	C₂₃×#463	—	2	1	no	—
48	C₂₃×#462	C₂²	1	1	yes	characteristic
49	C₂₃×#461	C₂²	1	1	yes	characteristic
50	C₂₃×#460	C₂²	1	1	yes	characteristic
51	C₂₃×#460	C₂²	1	1	yes	characteristic
52	C₂₃×#459	C₂²	1	1	yes	characteristic
53	C₂₃×#459	C₂²	1	1	yes	characteristic
54	C₂₃×#442	—	2	2	no	—
55	C₂₃×#423	C₂²	1	1	yes	characteristic
56	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
57	C₂₃×#416	—	2	1	no	—
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#331	C₂²	1	1	yes	characteristic
62	C₂₃×#331	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
69	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂×#41×C₂₃	—	2	1	no	—
73	C₂×#41×C₂₃	—	2	1	no	—
74	C₂×#41×C₂₃	—	2	1	no	—
75	C₂×#41×C₂₃	—	2	1	no	—
76	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
77	C₂₃×#135	—	4	4	no	—
78	C₂×Q₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	2	1	no	—
80	C₂×SD₁₆×C₂₃	—	2	1	no	—
81	C₂×SD₁₆×C₂₃	—	2	1	no	—
82	C₂×D₈×C₂₃	—	2	1	no	—
83	C₂×D₈×C₂₃	—	2	2	no	—
84	C₂×D₈×C₂₃	—	2	1	no	—
85	C₂×M₁₆×C₂₃	—	2	1	no	—
86	C₂×M₁₆×C₂₃	—	2	1	no	—
87	C₂²×C₈×C₂₃	—	2	1	no	—
88	C₂₃×C₄⋊Q₈	—	2	1	no	—
89	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₂₃×#127	—	2	2	no	—
92	C₂₃×#127	—	2	1	no	—
93	C₂₃×#127	—	2	1	no	—
94	C₂₃×#127	C₂³	1	1	yes	characteristic
95	C₂₃×#127	C₂³	1	1	yes	characteristic
96	C₂₃×#126	—	2	2	no	—
97	C₂₃×#126	—	2	1	no	—
98	C₂₃×#125	—	2	2	no	—
99	C₂₃×#125	—	2	1	no	—
100	C₂₃×#125	D₄	1	1	yes	characteristic
101	C₂₃×#124	—	2	1	no	—
102	C₂₃×#124	—	2	1	no	—
103	C₂₃×#124	D₄	1	1	yes	characteristic
104	C₂₃×#123	—	2	2	no	—
105	C₂₃×#123	—	2	2	no	—
106	C₂₃×#123	—	2	2	no	—
107	C₂₃×#123	—	2	1	no	—
108	C₂₃×#123	—	2	2	no	—
109	C₂₃×C₂²⋊Q₈	—	2	1	no	—
110	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
111	C₂₃×#121	—	2	2	no	—
112	C₂₃×#121	—	2	2	no	—
113	C₂₃×#121	—	2	2	no	—
114	C₂₃×#121	—	2	2	no	—
115	C₂₃×#121	—	2	2	no	—
116	C₂₃×#121	—	2	1	no	—
117	C₂₃×#121	—	2	2	no	—
118	C₂₃×#121	—	2	2	no	—
119	C₂₃×#121	—	2	1	no	—
120	C₂₃×#121	—	2	1	no	—
121	C₂₃×#121	—	2	1	no	—
122	C₂₃×#121	—	2	1	no	—
123	C₂₃×#121	—	2	1	no	—
124	C₂₃×#121	D₄	1	1	yes	characteristic
125	C₂₃×#121	D₄	1	2	yes	—
126	C₄×Q₈×C₂₃	—	2	1	no	—
127	C₄×Q₈×C₂₃	—	2	1	no	—
128	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
129	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
130	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
131	C₄×D₄×C₂₃	—	2	2	no	—
132	C₄×D₄×C₂₃	—	2	2	no	—
133	C₄×D₄×C₂₃	—	2	2	no	—
134	C₄×D₄×C₂₃	—	2	2	no	—
135	C₄×D₄×C₂₃	—	2	1	no	—
136	C₄×D₄×C₂₃	—	2	1	no	—
137	C₄×D₄×C₂₃	—	2	1	no	—
138	C₄×D₄×C₂₃	—	2	1	no	—
139	C₄×D₄×C₂₃	—	2	1	no	—
140	C₄×D₄×C₂₃	—	2	1	no	—
141	C₄×D₄×C₂₃	—	2	1	no	—
142	C₄×D₄×C₂₃	—	2	1	no	—
143	C₄×D₄×C₂₃	D₄	1	2	yes	—
144	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
145	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
146	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
147	C₂₃×#117	—	2	2	no	—
148	C₂₃×#117	—	2	1	no	—
149	C₂₃×#117	—	2	1	no	—
150	C₂₃×#117	D₄	1	1	yes	characteristic
151	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
153	C₂×C₄²×C₂₃	—	2	1	no	—
154	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
155	C₂₃×#107	—	2	1	no	—
156	C₂₃×#106	—	2	2	no	—
157	C₂₃×#106	—	2	1	no	—
158	C₂₃×C₄⋊C₈	—	2	1	no	—
159	C₂₃×C₄⋊C₈	—	2	1	no	—
160	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#103	—	2	1	no	—
167	C₂₃×#102	—	2	2	no	—
168	C₂₃×#102	—	2	1	no	—
169	C₂₃×#102	—	2	1	no	—
170	C₂₃×#102	—	2	1	no	—
171	C₂₃×#102	—	2	2	no	—
172	C₂₃×#102	—	2	1	no	—
173	C₂₃×#102	—	2	1	no	—
174	C₂₃×#97	—	2	1	no	—
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
177	#41×C₂₃	—	2	2	no	—
178	#41×C₂₃	—	4	1	no	—
179	#41×C₂₃	—	4	2	no	—
180	#41×C₂₃	—	4	2	no	—
181	#41×C₂₃	—	4	2	no	—
182	#41×C₂₃	—	4	2	no	—
183	#41×C₂₃	—	8	1	no	—
184	#41×C₂₃	—	8	1	no	—
185	#41×C₂₃	—	8	1	no	—
186	#41×C₂₃	—	8	1	no	—
187	#41×C₂₃	—	8	1	no	—
188	#41×C₂₃	—	8	1	no	—
189	C₂×Q₈×C₂₃	—	2	1	no	—
190	C₂×Q₈×C₂₃	—	2	1	no	—
191	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₂×D₄×C₂₃	—	2	2	no	—
194	C₂×D₄×C₂₃	—	2	1	no	—
195	C₂×D₄×C₂₃	—	2	2	no	—
196	C₂×D₄×C₂₃	—	2	2	no	—
197	C₂×D₄×C₂₃	—	2	2	no	—
198	C₂×D₄×C₂₃	—	2	2	no	—
199	C₂×D₄×C₂₃	—	2	1	no	—
200	C₂×D₄×C₂₃	—	2	2	no	—
201	C₂×D₄×C₂₃	—	2	1	no	—
202	C₂×D₄×C₂₃	—	2	1	no	—
203	C₂×D₄×C₂₃	—	2	1	no	—
204	C₂×D₄×C₂₃	—	2	1	no	—
205	C₂×D₄×C₂₃	—	2	1	no	—
206	C₂×D₄×C₂₃	—	4	1	no	—
207	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
208	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
209	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
210	C₂²×C₄×C₂₃	—	2	2	no	—
211	C₂²×C₄×C₂₃	—	2	2	no	—
212	C₂²×C₄×C₂₃	—	2	2	no	—
213	C₂²×C₄×C₂₃	—	2	1	no	—
214	C₂²×C₄×C₂₃	—	2	1	no	—
215	C₂²×C₄×C₂₃	—	2	1	no	—
216	C₂²×C₄×C₂₃	—	2	1	no	—
217	C₂²×C₄×C₂₃	—	2	2	no	—
218	C₂²×C₄×C₂₃	—	4	1	no	—
219	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
220	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
221	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
222	Q₁₆×C₂₃	—	4	2	no	—
223	SD₁₆×C₂₃	—	4	2	no	—
224	SD₁₆×C₂₃	—	4	2	no	—
225	SD₁₆×C₂₃	—	8	1	no	—
226	D₈×C₂₃	—	4	4	no	—
227	D₈×C₂₃	—	4	2	no	—
228	D₈×C₂₃	—	8	1	no	—
229	M₁₆×C₂₃	—	4	2	no	—
230	M₁₆×C₂₃	—	8	1	no	—
231	C₂×C₈×C₂₃	—	2	1	no	—
232	C₂×C₈×C₂₃	—	2	1	no	—
233	C₂×C₈×C₂₃	—	2	1	no	—
234	C₂×C₈×C₂₃	—	2	1	no	—
235	C₂×C₈×C₂₃	—	2	1	no	—
236	C₂×C₈×C₂₃	—	2	1	no	—
237	C₂×C₈×C₂₃	—	4	2	no	—
238	C₄⋊C₄×C₂₃	—	2	2	no	—
239	C₄⋊C₄×C₂₃	—	2	2	no	—
240	C₄⋊C₄×C₂₃	—	2	2	no	—
241	C₄⋊C₄×C₂₃	—	2	2	no	—
242	C₄⋊C₄×C₂₃	—	2	1	no	—
243	C₄⋊C₄×C₂₃	—	2	1	no	—
244	C₄⋊C₄×C₂₃	—	2	1	no	—
245	C₄⋊C₄×C₂₃	—	2	1	no	—
246	C₄⋊C₄×C₂₃	—	2	1	no	—
247	C₄⋊C₄×C₂₃	—	2	1	no	—
248	C₄⋊C₄×C₂₃	—	2	1	no	—
249	C₄⋊C₄×C₂₃	—	2	1	no	—
250	C₄⋊C₄×C₂₃	—	2	1	no	—
251	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
252	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
253	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
254	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
255	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
256	C₂²⋊C₄×C₂₃	—	2	2	no	—
257	C₂²⋊C₄×C₂₃	—	2	2	no	—
258	C₂²⋊C₄×C₂₃	—	2	2	no	—
259	C₂²⋊C₄×C₂₃	—	2	1	no	—
260	C₂²⋊C₄×C₂₃	—	2	1	no	—
261	C₂²⋊C₄×C₂₃	—	2	1	no	—
262	C₂²⋊C₄×C₂₃	—	2	1	no	—
263	C₂²⋊C₄×C₂₃	—	2	2	no	—
264	C₂²⋊C₄×C₂₃	—	2	1	no	—
265	C₂²⋊C₄×C₂₃	—	2	1	no	—
266	C₂²⋊C₄×C₂₃	—	2	2	no	—
267	C₂²⋊C₄×C₂₃	—	2	1	no	—
268	C₂²⋊C₄×C₂₃	—	2	1	no	—
269	C₂²⋊C₄×C₂₃	—	4	1	no	—
270	C₂²⋊C₄×C₂₃	—	4	1	no	—
271	C₄²×C₂₃	—	2	2	no	—
272	C₄²×C₂₃	—	2	1	no	—
273	C₄²×C₂₃	—	2	1	no	—
274	C₄²×C₂₃	—	2	1	no	—
275	C₄²×C₂₃	—	2	1	no	—
276	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
277	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
278	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
279	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
280	#18606	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
281	C₂³×C₂₃	—	2	1	no	—
282	C₂³×C₂₃	—	2	1	no	—
283	C₂³×C₂₃	—	2	2	no	—
284	C₂³×C₂₃	—	4	1	no	—
285	C₂³×C₂₃	#120	1	1	yes	characteristic
286	Q₈×C₂₃	—	2	1	no	—
287	Q₈×C₂₃	—	2	1	no	—
288	Q₈×C₂₃	—	2	1	no	—
289	Q₈×C₂₃	—	4	1	no	—
290	Q₈×C₂₃	—	4	1	no	—
291	Q₈×C₂₃	—	4	1	no	—
292	Q₈×C₂₃	—	4	1	no	—
293	Q₈×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
294	D₄×C₂₃	—	2	1	no	—
295	D₄×C₂₃	—	2	1	no	—
296	D₄×C₂₃	—	2	1	no	—
297	D₄×C₂₃	—	2	1	no	—
298	D₄×C₂₃	—	4	2	no	—
299	D₄×C₂₃	—	4	2	no	—
300	D₄×C₂₃	—	4	4	no	—
301	D₄×C₂₃	—	4	2	no	—
302	D₄×C₂₃	—	4	1	no	—
303	D₄×C₂₃	—	4	1	no	—
304	D₄×C₂₃	—	4	1	no	—
305	D₄×C₂₃	—	4	1	no	—
306	D₄×C₂₃	—	4	1	no	—
307	D₄×C₂₃	—	4	1	no	—
308	D₄×C₂₃	—	4	1	no	—
309	D₄×C₂₃	—	4	1	no	—
310	D₄×C₂₃	—	8	1	no	—
311	D₄×C₂₃	—	8	2	no	—
312	D₄×C₂₃	—	8	2	no	—
313	D₄×C₂₃	—	8	2	no	—
314	D₄×C₂₃	—	8	2	no	—
315	D₄×C₂₃	—	8	2	no	—
316	D₄×C₂₃	—	8	2	no	—
317	C₂×C₄×C₂₃	—	2	1	no	—
318	C₂×C₄×C₂₃	—	2	1	no	—
319	C₂×C₄×C₂₃	—	2	1	no	—
320	C₂×C₄×C₂₃	—	2	1	no	—
321	C₂×C₄×C₂₃	—	2	1	no	—
322	C₂×C₄×C₂₃	—	2	1	no	—
323	C₂×C₄×C₂₃	—	2	1	no	—
324	C₂×C₄×C₂₃	—	2	1	no	—
325	C₂×C₄×C₂₃	—	2	1	no	—
326	C₂×C₄×C₂₃	—	2	2	no	—
327	C₂×C₄×C₂₃	—	2	1	no	—
328	C₂×C₄×C₂₃	—	4	1	no	—
329	C₂×C₄×C₂₃	—	4	2	no	—
330	C₂×C₄×C₂₃	—	4	4	no	—
331	C₂×C₄×C₂₃	—	4	1	no	—
332	C₂×C₄×C₂₃	—	4	1	no	—
333	C₂×C₄×C₂₃	—	4	1	no	—
334	C₂×C₄×C₂₃	—	4	1	no	—
335	C₂×C₄×C₂₃	—	4	1	no	—
336	C₂×C₄×C₂₃	—	8	1	no	—
337	C₂×C₄×C₂₃	—	8	2	no	—
338	C₂×C₄×C₂₃	—	8	2	no	—
339	C₂×C₄×C₂₃	—	8	2	no	—
340	C₂×C₄×C₂₃	—	8	2	no	—
341	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
342	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
343	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
344	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
345	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
346	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
347	C₈×C₂₃	—	2	2	no	—
348	C₈×C₂₃	—	4	1	no	—
349	C₈×C₂₃	—	4	1	no	—
350	C₈×C₂₃	—	4	1	no	—
351	C₈×C₂₃	—	4	1	no	—
352	C₈×C₂₃	—	4	1	no	—
353	#3517	C₂×C₂₃	1	1	yes	characteristic, complemented
354	#3251	C₂×C₂₃	1	1	yes	characteristic, complemented
355	#3184	C₂×C₂₃	1	1	yes	characteristic, complemented
356	#3035	C₂×C₂₃	1	1	yes	characteristic, complemented
357	#1654	C₂×C₂₃	1	1	yes	characteristic, complemented
358	#1653	C₂×C₂₃	1	1	yes	characteristic, complemented
359	#1642	C₂×C₂₃	1	1	yes	characteristic, complemented
360	#1641	C₂×C₂₃	1	1	yes	characteristic, complemented
361	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
362	#1625	C₂×C₂₃	1	1	yes	characteristic, complemented
363	#1621	C₂×C₂₃	1	1	yes	characteristic, complemented
364	#1621	C₂×C₂₃	1	1	yes	characteristic, complemented
365	#1547	C₂×C₂₃	1	1	yes	characteristic, complemented
366	#1489	C₂×C₂₃	1	1	yes	characteristic, complemented
367	#1483	C₂×C₂₃	1	1	yes	characteristic, complemented
368	C₂²×C₂₃	—	2	1	no	—
369	C₂²×C₂₃	—	2	1	no	—
370	C₂²×C₂₃	—	2	1	no	—
371	C₂²×C₂₃	—	4	1	no	—
372	C₂²×C₂₃	—	4	2	no	—
373	C₂²×C₂₃	—	4	1	no	—
374	C₂²×C₂₃	—	4	1	no	—
375	C₂²×C₂₃	—	4	1	no	—
376	C₂²×C₂₃	—	4	2	no	—
377	C₂²×C₂₃	—	4	1	no	—
378	C₂²×C₂₃	—	4	2	no	—
379	C₂²×C₂₃	—	4	1	no	—
380	C₂²×C₂₃	—	8	1	no	—
381	C₂²×C₂₃	—	8	1	no	—
382	C₂²×C₂₃	—	8	1	no	—
383	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
384	C₄×C₂₃	—	2	1	no	—
385	C₄×C₂₃	—	2	2	no	—
386	C₄×C₂₃	—	2	1	no	—
387	C₄×C₂₃	—	2	1	no	—
388	C₄×C₂₃	—	2	1	no	—
389	C₄×C₂₃	—	4	1	no	—
390	C₄×C₂₃	—	4	1	no	—
391	C₄×C₂₃	—	4	1	no	—
392	C₄×C₂₃	—	4	2	no	—
393	C₄×C₂₃	—	4	1	no	—
394	C₄×C₂₃	—	4	1	no	—
395	C₄×C₂₃	—	8	1	no	—
396	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
397	C₄×C₂₃	#575	1	1	yes	characteristic
398	C₄×C₂₃	#576	1	1	yes	characteristic
399	C₄×C₂₃	#577	1	1	yes	characteristic
400	C₂×#135	—	2	1	no	—
401	#557	C₂²×C₂₃	1	1	yes	characteristic
402	#553	—	2	2	no	—
403	#553	—	2	1	no	—
404	#550	—	2	1	no	—
405	#550	—	2	1	no	—
406	#550	C₂²×C₂₃	1	1	yes	characteristic
407	#550	C₂²×C₂₃	1	1	yes	characteristic
408	#547	—	2	1	no	—
409	#547	C₂²×C₂₃	1	1	yes	characteristic
410	#540	—	2	2	no	—
411	#540	—	2	1	no	—
412	#532	—	2	1	no	—
413	#532	C₂²×C₂₃	1	1	yes	characteristic
414	#532	C₂²×C₂₃	1	1	yes	characteristic
415	#529	—	2	1	no	—
416	#529	C₂²×C₂₃	1	1	yes	characteristic
417	#520	—	2	1	no	—
418	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
419	#498	C₂²×C₂₃	1	1	yes	characteristic
420	#493	C₂²×C₂₃	1	1	yes	characteristic
421	#486	C₂²×C₂₃	1	1	yes	characteristic
422	#486	C₂²×C₂₃	1	1	yes	characteristic
423	#478	—	2	1	no	—
424	#475	C₂²×C₂₃	1	1	yes	characteristic
425	#474	C₂²×C₂₃	1	1	yes	characteristic
426	#469	—	2	2	no	—
427	#469	—	2	1	no	—
428	#468	—	2	1	no	—
429	#464	—	2	1	no	—
430	#463	—	2	1	no	—
431	#463	—	2	1	no	—
432	#462	C₂²×C₂₃	1	1	yes	characteristic
433	#461	C₂²×C₂₃	1	1	yes	characteristic
434	#460	C₂²×C₂₃	1	1	yes	characteristic
435	#460	C₂²×C₂₃	1	1	yes	characteristic
436	#459	C₂²×C₂₃	1	1	yes	characteristic
437	#459	C₂²×C₂₃	1	1	yes	characteristic
438	#442	—	2	2	no	—
439	#423	C₂²×C₂₃	1	1	yes	characteristic
440	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
441	#416	—	2	1	no	—
442	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
443	#332	C₂²×C₂₃	1	1	yes	characteristic
444	#332	C₂²×C₂₃	1	1	yes	characteristic
445	#331	C₂²×C₂₃	1	1	yes	characteristic
446	#331	C₂²×C₂₃	1	1	yes	characteristic
447	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
448	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
449	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
450	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
451	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
452	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
453	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
454	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
455	C₂×C₂₃	—	4	1	no	—
456	C₂×C₂₃	—	8	1	no	—
457	C₂×C₂₃	—	8	1	no	—
458	C₂×C₂₃	—	8	2	no	—
459	C₂×C₂₃	—	16	1	no	—
460	C₂×C₂₃	#3006	1	1	yes	characteristic
461	C₂×C₂₃	#3012	1	1	yes	characteristic
462	C₂×C₂₃	#3018	1	1	yes	characteristic
463	C₂×#41	—	2	1	no	—
464	C₂×#41	—	2	1	no	—
465	C₂×#41	—	2	1	no	—
466	C₂×#41	—	2	1	no	—
467	C₂×#41	—	2	1	no	—
468	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
469	#135	—	4	4	no	—
470	C₂×Q₁₆	—	2	1	no	—
471	C₂×SD₁₆	—	2	1	no	—
472	C₂×SD₁₆	—	2	1	no	—
473	C₂×SD₁₆	—	2	1	no	—
474	C₂×D₈	—	2	1	no	—
475	C₂×D₈	—	2	2	no	—
476	C₂×D₈	—	2	1	no	—
477	C₂×M₁₆	—	2	1	no	—
478	C₂×M₁₆	—	2	1	no	—
479	C₂²×C₈	—	2	1	no	—
480	C₄⋊Q₈	—	2	1	no	—
481	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
482	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
483	#127	—	2	2	no	—
484	#127	—	2	1	no	—
485	#127	—	2	1	no	—
486	#127	C₂³×C₂₃	1	1	yes	characteristic
487	#127	C₂³×C₂₃	1	1	yes	characteristic
488	#126	—	2	2	no	—
489	#126	—	2	1	no	—
490	#125	—	2	2	no	—
491	#125	—	2	1	no	—
492	#125	D₄×C₂₃	1	1	yes	characteristic
493	#124	—	2	1	no	—
494	#124	—	2	1	no	—
495	#124	D₄×C₂₃	1	1	yes	characteristic
496	#123	—	2	2	no	—
497	#123	—	2	2	no	—
498	#123	—	2	2	no	—
499	#123	—	2	1	no	—
500	#123	—	2	2	no	—
501	C₂²⋊Q₈	—	2	1	no	—
502	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
503	#121	—	2	2	no	—
504	#121	—	2	2	no	—
505	#121	—	2	2	no	—
506	#121	—	2	2	no	—
507	#121	—	2	2	no	—
508	#121	—	2	1	no	—
509	#121	—	2	2	no	—
510	#121	—	2	2	no	—
511	#121	—	2	1	no	—
512	#121	—	2	1	no	—
513	#121	—	2	1	no	—
514	#121	—	2	1	no	—
515	#121	—	2	1	no	—
516	#121	D₄×C₂₃	1	1	yes	characteristic
517	#121	D₄×C₂₃	1	2	yes	—
518	C₄×Q₈	—	2	1	no	—
519	C₄×Q₈	—	2	1	no	—
520	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
521	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
522	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
523	C₄×D₄	—	2	2	no	—
524	C₄×D₄	—	2	2	no	—
525	C₄×D₄	—	2	2	no	—
526	C₄×D₄	—	2	2	no	—
527	C₄×D₄	—	2	1	no	—
528	C₄×D₄	—	2	1	no	—
529	C₄×D₄	—	2	1	no	—
530	C₄×D₄	—	2	1	no	—
531	C₄×D₄	—	2	1	no	—
532	C₄×D₄	—	2	1	no	—
533	C₄×D₄	—	2	1	no	—
534	C₄×D₄	—	2	1	no	—
535	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
536	C₄×D₄	D₄×C₂₃	1	2	yes	—
537	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
538	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
539	#117	—	2	2	no	—
540	#117	—	2	1	no	—
541	#117	—	2	1	no	—
542	#117	D₄×C₂₃	1	1	yes	characteristic
543	C₂×C₄⋊C₄	—	2	2	no	—
544	C₂×C₄⋊C₄	—	2	1	no	—
545	C₂×C₄²	—	2	1	no	—
546	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
547	#107	—	2	1	no	—
548	#106	—	2	2	no	—
549	#106	—	2	1	no	—
550	C₄⋊C₈	—	2	1	no	—
551	C₄⋊C₈	—	2	1	no	—
552	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
553	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
554	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
555	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
556	#103	—	2	1	no	—
557	#103	—	2	1	no	—
558	#103	—	2	1	no	—
559	#102	—	2	2	no	—
560	#102	—	2	1	no	—
561	#102	—	2	1	no	—
562	#102	—	2	1	no	—
563	#102	—	2	2	no	—
564	#102	—	2	1	no	—
565	#102	—	2	1	no	—
566	#97	—	2	1	no	—
567	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
568	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
569	C₂₃	#18606	1	1	yes	characteristic, direct product factor, Sylow
570	#41	—	2	2	no	—
571	#41	—	4	1	no	—
572	#41	—	4	2	no	—
573	#41	—	4	2	no	—
574	#41	—	4	2	no	—
575	#41	—	4	2	no	—
576	#41	—	8	1	no	—
577	#41	—	8	1	no	—
578	#41	—	8	1	no	—
579	#41	—	8	1	no	—
580	#41	—	8	1	no	—
581	#41	—	8	1	no	—
582	C₂×Q₈	—	2	1	no	—
583	C₂×Q₈	—	2	1	no	—
584	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
585	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
586	C₂×D₄	—	2	2	no	—
587	C₂×D₄	—	2	1	no	—
588	C₂×D₄	—	2	2	no	—
589	C₂×D₄	—	2	2	no	—
590	C₂×D₄	—	2	2	no	—
591	C₂×D₄	—	2	2	no	—
592	C₂×D₄	—	2	1	no	—
593	C₂×D₄	—	2	2	no	—
594	C₂×D₄	—	2	1	no	—
595	C₂×D₄	—	2	1	no	—
596	C₂×D₄	—	2	1	no	—
597	C₂×D₄	—	2	1	no	—
598	C₂×D₄	—	2	1	no	—
599	C₂×D₄	—	4	1	no	—
600	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
601	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
602	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
603	C₂²×C₄	—	2	2	no	—
604	C₂²×C₄	—	2	2	no	—
605	C₂²×C₄	—	2	2	no	—
606	C₂²×C₄	—	2	1	no	—
607	C₂²×C₄	—	2	1	no	—
608	C₂²×C₄	—	2	1	no	—
609	C₂²×C₄	—	2	1	no	—
610	C₂²×C₄	—	2	2	no	—
611	C₂²×C₄	—	4	1	no	—
612	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
613	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
614	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
615	Q₁₆	—	4	2	no	—
616	SD₁₆	—	4	2	no	—
617	SD₁₆	—	4	2	no	—
618	SD₁₆	—	8	1	no	—
619	D₈	—	4	4	no	—
620	D₈	—	4	2	no	—
621	D₈	—	8	1	no	—
622	M₁₆	—	4	2	no	—
623	M₁₆	—	8	1	no	—
624	C₂×C₈	—	2	1	no	—
625	C₂×C₈	—	2	1	no	—
626	C₂×C₈	—	2	1	no	—
627	C₂×C₈	—	2	1	no	—
628	C₂×C₈	—	2	1	no	—
629	C₂×C₈	—	2	1	no	—
630	C₂×C₈	—	4	2	no	—
631	C₄⋊C₄	—	2	2	no	—
632	C₄⋊C₄	—	2	2	no	—
633	C₄⋊C₄	—	2	2	no	—
634	C₄⋊C₄	—	2	2	no	—
635	C₄⋊C₄	—	2	1	no	—
636	C₄⋊C₄	—	2	1	no	—
637	C₄⋊C₄	—	2	1	no	—
638	C₄⋊C₄	—	2	1	no	—
639	C₄⋊C₄	—	2	1	no	—
640	C₄⋊C₄	—	2	1	no	—
641	C₄⋊C₄	—	2	1	no	—
642	C₄⋊C₄	—	2	1	no	—
643	C₄⋊C₄	—	2	1	no	—
644	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
645	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
646	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
647	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
648	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
649	C₂²⋊C₄	—	2	2	no	—
650	C₂²⋊C₄	—	2	2	no	—
651	C₂²⋊C₄	—	2	2	no	—
652	C₂²⋊C₄	—	2	1	no	—
653	C₂²⋊C₄	—	2	1	no	—
654	C₂²⋊C₄	—	2	1	no	—
655	C₂²⋊C₄	—	2	1	no	—
656	C₂²⋊C₄	—	2	2	no	—
657	C₂²⋊C₄	—	2	1	no	—
658	C₂²⋊C₄	—	2	1	no	—
659	C₂²⋊C₄	—	2	2	no	—
660	C₂²⋊C₄	—	2	1	no	—
661	C₂²⋊C₄	—	2	1	no	—
662	C₂²⋊C₄	—	4	1	no	—
663	C₂²⋊C₄	—	4	1	no	—
664	C₄²	—	2	2	no	—
665	C₄²	—	2	1	no	—
666	C₄²	—	2	1	no	—
667	C₄²	—	2	1	no	—
668	C₄²	—	2	1	no	—
669	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
670	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
671	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
672	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
673	C₂³	—	2	1	no	—
674	C₂³	—	2	1	no	—
675	C₂³	—	2	2	no	—
676	C₂³	—	4	1	no	—
677	C₂³	C₂₃×#120	1	1	yes	characteristic
678	Q₈	—	2	1	no	—
679	Q₈	—	2	1	no	—
680	Q₈	—	2	1	no	—
681	Q₈	—	4	1	no	—
682	Q₈	—	4	1	no	—
683	Q₈	—	4	1	no	—
684	Q₈	—	4	1	no	—
685	Q₈	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
686	D₄	—	2	1	no	—
687	D₄	—	2	1	no	—
688	D₄	—	2	1	no	—
689	D₄	—	2	1	no	—
690	D₄	—	4	2	no	—
691	D₄	—	4	2	no	—
692	D₄	—	4	4	no	—
693	D₄	—	4	2	no	—
694	D₄	—	4	1	no	—
695	D₄	—	4	1	no	—
696	D₄	—	4	1	no	—
697	D₄	—	4	1	no	—
698	D₄	—	4	1	no	—
699	D₄	—	4	1	no	—
700	D₄	—	4	1	no	—
701	D₄	—	4	1	no	—
702	D₄	—	8	2	no	—
703	D₄	—	8	1	no	—
704	D₄	—	8	2	no	—
705	D₄	—	8	2	no	—
706	D₄	—	8	2	no	—
707	D₄	—	8	2	no	—
708	D₄	—	8	2	no	—
709	C₂×C₄	—	2	1	no	—
710	C₂×C₄	—	2	1	no	—
711	C₂×C₄	—	2	1	no	—
712	C₂×C₄	—	2	1	no	—
713	C₂×C₄	—	2	1	no	—
714	C₂×C₄	—	2	1	no	—
715	C₂×C₄	—	2	1	no	—
716	C₂×C₄	—	2	1	no	—
717	C₂×C₄	—	2	1	no	—
718	C₂×C₄	—	2	2	no	—
719	C₂×C₄	—	2	1	no	—
720	C₂×C₄	—	4	1	no	—
721	C₂×C₄	—	4	2	no	—
722	C₂×C₄	—	4	4	no	—
723	C₂×C₄	—	4	1	no	—
724	C₂×C₄	—	4	1	no	—
725	C₂×C₄	—	4	1	no	—
726	C₂×C₄	—	4	1	no	—
727	C₂×C₄	—	4	1	no	—
728	C₂×C₄	—	8	2	no	—
729	C₂×C₄	—	8	1	no	—
730	C₂×C₄	—	8	2	no	—
731	C₂×C₄	—	8	2	no	—
732	C₂×C₄	—	8	2	no	—
733	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
734	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
735	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
736	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
737	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
738	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
739	C₈	—	2	2	no	—
740	C₈	—	4	1	no	—
741	C₈	—	4	1	no	—
742	C₈	—	4	1	no	—
743	C₈	—	4	1	no	—
744	C₈	—	4	1	no	—
745	C₂²	—	2	1	no	—
746	C₂²	—	2	1	no	—
747	C₂²	—	2	1	no	—
748	C₂²	—	4	1	no	—
749	C₂²	—	4	2	no	—
750	C₂²	—	4	1	no	—
751	C₂²	—	4	1	no	—
752	C₂²	—	4	1	no	—
753	C₂²	—	4	2	no	—
754	C₂²	—	4	1	no	—
755	C₂²	—	4	2	no	—
756	C₂²	—	4	1	no	—
757	C₂²	—	8	1	no	—
758	C₂²	—	8	1	no	—
759	C₂²	—	8	1	no	—
760	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
761	C₄	—	2	1	no	—
762	C₄	—	2	2	no	—
763	C₄	—	2	1	no	—
764	C₄	—	2	1	no	—
765	C₄	—	2	1	no	—
766	C₄	—	4	1	no	—
767	C₄	—	4	1	no	—
768	C₄	—	4	1	no	—
769	C₄	—	4	2	no	—
770	C₄	—	4	1	no	—
771	C₄	—	4	1	no	—
772	C₄	—	8	1	no	—
773	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
774	C₄	C₂₃×#575	1	1	yes	characteristic
775	C₄	C₂₃×#576	1	1	yes	characteristic
776	C₄	C₂₃×#577	1	1	yes	characteristic
777	C₂	—	4	1	no	—
778	C₂	—	8	1	no	—
779	C₂	—	8	1	no	—
780	C₂	—	8	2	no	—
781	C₂	—	16	1	no	—
782	C₂	C₂₃×#3006	1	1	yes	characteristic
783	C₂	C₂₃×#3012	1	1	yes	characteristic
784	C₂	C₂₃×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#3517	C₂	semidirect
2. C₂₃×#3251	C₂	semidirect
3. C₂₃×#3184	C₂	semidirect
4. C₂₃×#3035	C₂	semidirect
5. C₂₃×#1654	C₂	semidirect
6. C₂₃×#1653	C₂	semidirect
7. C₂₃×#1642	C₂	semidirect
8. C₂₃×#1641	C₂	semidirect
9. C₂₃×#1626	C₂	semidirect
10. C₂₃×#1625	C₂	semidirect
11. C₂₃×#1621	C₂	semidirect
12. C₂₃×#1621	C₂	semidirect
13. C₂₃×#1547	C₂	semidirect
14. C₂₃×#1489	C₂	semidirect
15. C₂₃×#1483	C₂	semidirect
280. #18606	C₂₃	direct (N × H)
353. #3517	C₂×C₂₃	semidirect
354. #3251	C₂×C₂₃	semidirect
355. #3184	C₂×C₂₃	semidirect
356. #3035	C₂×C₂₃	semidirect
357. #1654	C₂×C₂₃	semidirect
358. #1653	C₂×C₂₃	semidirect
359. #1642	C₂×C₂₃	semidirect
360. #1641	C₂×C₂₃	semidirect
361. #1626	C₂×C₂₃	semidirect
362. #1625	C₂×C₂₃	semidirect
363. #1621	C₂×C₂₃	semidirect
364. #1621	C₂×C₂₃	semidirect
365. #1547	C₂×C₂₃	semidirect
366. #1489	C₂×C₂₃	semidirect
367. #1483	C₂×C₂₃	semidirect
569. C₂₃	#18606	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3517	2
C₂₃×#3251	2
C₂₃×#3184	2
C₂₃×#3035	2
C₂₃×#1654	2
C₂₃×#1653	2
C₂₃×#1642	2
C₂₃×#1641	2
C₂₃×#1626	2
C₂₃×#1625	2
C₂₃×#1621	2
C₂₃×#1547	2
C₂₃×#1489	2
C₂₃×#1483	2
#18606	23

Subgroups — C23×#18606 #19166232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18606 #19166232